And just as in English we don't "write" passwords but type them, in Hebrew you would usually use the verb lehaklid, which comes from mikleded - keyboard:
אני מקלידה את הסיסמא
The original intent of this sentence was not related to a digital password. What gave you this indication?
What other kind is there in the modern world?
But yes, I can see how the sentence could mean that she's writing it down on a piece of paper so that she doesn't forget it. In that case kotevet is fine.
Great question :). Short answer: both are correct. Long answer: the origin of this word is from the Aramaic language. In Aramaic א in the end of the word marks the definite article. Many words in Hebrew came from Aramaic with this א suffix and mistakenly though to be feminine (indefinite) words. A few years ago the Academy of the Hebrew Language decided that nouns in Hebrew will never and with א. All of the nouns that already ends with א, could be written both with א or ה.
You will write אם or אמי/אמך/אמנו/וכדומה. :) the word אמא is more like a nickname than a noun.